Once again, both products are easily evaluated (See Ref.[1]}):
\(\begin{split}\log \triangle \triangle &=-\frac{i k^2}{4}\sum \limits _{n=-\infty }^{+\infty }\frac{1}{\alpha n^2-\beta +\gamma n}\\&=\frac{i\pi k^2}{4\sqrt{\gamma ^2+4\alpha \beta }}\left[\cot {\pi a_{+}}-\cot {\pi a_{-}}\right]\end{split}\) 
